| L(s) = 1 | − 3-s + 3·7-s − 2·9-s + 6·11-s + 6·13-s + 2·19-s − 3·21-s − 23-s + 5·27-s − 2·29-s − 10·31-s − 6·33-s + 10·37-s − 6·39-s + 9·41-s + 4·43-s + 2·49-s − 2·57-s + 8·59-s − 6·61-s − 6·63-s + 5·67-s + 69-s + 6·71-s − 10·73-s + 18·77-s + 2·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s − 2/3·9-s + 1.80·11-s + 1.66·13-s + 0.458·19-s − 0.654·21-s − 0.208·23-s + 0.962·27-s − 0.371·29-s − 1.79·31-s − 1.04·33-s + 1.64·37-s − 0.960·39-s + 1.40·41-s + 0.609·43-s + 2/7·49-s − 0.264·57-s + 1.04·59-s − 0.768·61-s − 0.755·63-s + 0.610·67-s + 0.120·69-s + 0.712·71-s − 1.17·73-s + 2.05·77-s + 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.936100892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.936100892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.86307764760211, −14.67221519066932, −14.25564913355445, −13.69986572151202, −13.06745069175473, −12.38056172970398, −11.80207475525295, −11.28393914346600, −11.13228193078969, −10.69393079799760, −9.591208706821813, −9.155821609371883, −8.706383609783602, −8.088020610038585, −7.509197956718606, −6.739846452335789, −6.098838498050252, −5.787030114023512, −5.140677648097486, −4.187043409636971, −3.953246379932679, −3.120617836698021, −2.051593557182222, −1.318306535145979, −0.7868086373667522,
0.7868086373667522, 1.318306535145979, 2.051593557182222, 3.120617836698021, 3.953246379932679, 4.187043409636971, 5.140677648097486, 5.787030114023512, 6.098838498050252, 6.739846452335789, 7.509197956718606, 8.088020610038585, 8.706383609783602, 9.155821609371883, 9.591208706821813, 10.69393079799760, 11.13228193078969, 11.28393914346600, 11.80207475525295, 12.38056172970398, 13.06745069175473, 13.69986572151202, 14.25564913355445, 14.67221519066932, 14.86307764760211
